Questions tagged [linear-pde]

Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.

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Question about the second order linear elliptic PDE on closed manifold

Recently I see a question linear second order PDE in which user Pedro post a reference in Gilbarg's book, which said that the solvability of the linear PDE $$ \Delta u +B^{i}(x)u_{i}+C(x)u=f $$ is ...
Elio Li's user avatar
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Riesz transform after linear transformation

I am encountering the term $\partial_x \mathcal{R}_x(f(x,y))$. I needed to do the following linear transformation $$x' = a x+ by,\,\,\,\,\, y'=ax-by,\,\,\, and \,\,f(x,y)=g(x',y') $$ I ended up with ...
Mr. Proof's user avatar
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Estimates for higher order derivatives of the Airy Kernel

Consider the kdv equation (from here) $$\left\{\begin{array}{l} \partial_{t} v+\partial_{x}^{3} v=0 \\ v(x, 0)=v_{0}(x) \end{array}\right.$$ Its solution can be written as $v(t,x)=S_t*v_0(x),$ where $...
Student's user avatar
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2 answers
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Under which conditions does this PDE have unique solutions

Assume that $f:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ is smooth and consider the linear equation $$\mathrm{div}\, (u)(x) = f(x,u(x)),$$ where $u:\mathbb{R}^n \to \mathbb{R}^n $ is a smooth ...
mlainz's user avatar
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4 votes
1 answer
218 views

Asymptotics of integral representation of distribution

I initially posted this question at MSE (here), but I have gotten no response, so I figured I would ask it to this community. Background: I am studying the PDE $$\,\,\,\,\,\,\,\,\,\,\,\,i\partial_t \...
Dispersion's user avatar
1 vote
1 answer
379 views

Explicit solution for a linear drift-diffusion equation (Fokker-Planck equation) on whole space

I'm wondering if there might be an explicit solution for the following linear PDE in two space dimensions $(x_1,x_2)$ on the whole space $\mathbb{R}^2$: $$ \partial_t f = {div} \left [\left( \...
kumquat's user avatar
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Uniqueness of the "weak solution" to Fokker-Plank PDE

Let $C_b^2(\mathbb R_+)$ be the set of functions $f: \mathbb R_+\to\mathbb R$ s.t. $f, f' ,f''$ are bounded and $f(0)=0$. Consider a measurable function $p: \mathbb R_+^2\to\mathbb R_+$ satisfying $$\...
GJC20's user avatar
  • 1,220
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1 answer
162 views

Explicit solutions for linear system of PDEs with constant coefficients

I've been recently trying to to solve the following system of linear 1st order PDE's: $f:\Omega^d\xrightarrow{}\mathbb{R},\quad A^{(k)}\in\mathbb{R}^{N\times N},\quad B^{(k)}\in\mathbb{R}^N$ $\dfrac{\...
IdoAmos's user avatar
1 vote
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Parabolic theory for singular coefficients on bounded domains (Reference Request)

In Evans, the theory for linear PDE of parabolic type with bounded coefficients is developed. There are nice results such as long-existence of weak solutions and the parabolic regularity theorems. Is ...
valcofadden's user avatar
2 votes
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Using Darboux's to solve 2D system of first order linear PDEs with variable coefficients

I've spent some time over the last few days looking at the references suggested in this question and this question and I think the information therein is my best shot at solving this system that arose ...
Leif Ericson's user avatar
1 vote
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138 views

Generalized functional for solution of PDEs

Asked this on Math Stack Exchange awhile ago but it got ignored then deleted. To solve a differential equation of one variable, you need constraints equal to the number of derivatives. For a partial ...
Connor Dolan's user avatar
3 votes
1 answer
318 views

Maximum principle and linear transport

Let us consider the linear transport equation $$ \partial_t u + \mathrm{div}(a(t,x)u)=0 $$ with initial data $u(0,\cdot) = u_0$ in $\mathbb R^N$. Here we consider a smooth Lipschitz vector field $a$. ...
Riku's user avatar
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1 answer
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Spectral analysis for nonlocal elliptic operator

Suppose $\Omega\subset\mathbb{R}^3$ is a bounded domain with smooth boundary. We note by $(-\Delta)^{-1}$ the inverse Laplacian i.e. $f\mapsto u$ where $u$ is the unique solution to $$-\Delta u=f,\...
Fozz's user avatar
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Liouville theorem for an elliptic equation with gradient perturbation

How can I prove the following Liouville theorem for an elliptic equation with gradient perturbation? Let $u \in L^2(\mathbb R^n;\mathbb R)$ be a smooth solution of $$ -\Delta u + v \cdot \nabla u = 0 ...
Lao's user avatar
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Can we find a uniform bound of the solution of a series of linear partial differential equations related to a parameter

Let $\sigma \in[0,1]$,we consider following series of linear partial differential equations related to the parameter $\sigma$,for example $$ \left\{\begin{aligned} \Delta \Phi &=\sigma f(x, y) \...
Elio Li's user avatar
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0 answers
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Decay of solution for linear system with damping

Let us consider the following linear system with damping: $$ \begin{cases} u_t - u_x = -\frac{1}{2} (u+v)\\ v_t + v_x = -\frac{1}{2} (u+v) \end{cases} $$ Let's write the solution as $w=(u,v)$ ...
Riku's user avatar
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0 answers
112 views

Solve a coupled PDE in a rectangle

We consider a coupled PDE in a rectangle $\Omega=(-1,1)\times(-1,1)$. For the simplicity, we assume that the functions are periodic in $x_{1}$ direction. \begin{cases} \nabla\cdot u=f_{1},\ & \...
Kira Yamato's user avatar
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Regularity of solution to Cauchy problem given regular initial data

Let $f\in L^2_1([0,T]\times \mathbb{T}^m)$ (Sobolev space of maps of regularity $1$, $\mathbb{T}^m$ is the $m$-dimensional torus) be a solution of a Cauchy problem $$\frac{d}{dt} f(t) = A f(t)$$ $$f(0)...
Overflowian's user avatar
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Reference for global theory of Schrödinger operators

Question. What is a good reference to learn about the spectral properties of Schrödinger operators in $\mathbf{R}^n$? I am specifically interested in references that discuss examples where the ...
Leo Moos's user avatar
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11 votes
2 answers
632 views

A singular differential equation

In a neighbourhood of $0$ in $\mathbb{R}^n$ a smooth function $h=h(x)$, $h(0)=0$, is given. Take arbitrary real numbers $w,\lambda_1,\dots,\lambda_n\in\mathbb{R}$. The problem is to find a smooth ...
Janusz's user avatar
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Has this form of the heat equation been solved for the radiation boundary condition

Below is a solution to a special form of the heat equation. I have found the postings on the heat equation and they are far above my head. I tried to find a tag on Transport Theory for both heat and ...
Michael Bernthal's user avatar
2 votes
2 answers
124 views

Density of traces of solutions to an elliptic equation

Let $D_1$ be a domain with smooth boundary and assume that $D_1$ is a proper subset of $D_2$ which is itself a bounded domain in $\mathbb R^n$ with a smooth boundary. Assume also that $D_2\setminus ...
Ali's user avatar
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3 votes
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How to find a particular solution of a non-homogeneous parabolic partial differential equation

Consider the following non-homogeneous parabolic partial differential equation (PDE) \begin{align} \left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma}{r} \sin\psi \frac{\partial}{\partial \...
GilbertDu's user avatar
4 votes
2 answers
248 views

spaces of smooth functions for linear hyperbolic PDE

Which classes of (scalar or systems of) linear first or second order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that there is a retarded Green's function $G:...
Arnold Neumaier's user avatar
2 votes
1 answer
143 views

Existence of a global analytic solution to a linear first order PDE

Let $B=\lbrace \|z\|<1\rbrace$ be a unit ball in $\mathbb{C}^n, n\geq 2.$ Let $f_1,\cdots, f_n, f$ be holomorphic functions on $B.$ Now, consider the following first order, linear PDE: $$f_1\...
John Z.'s user avatar
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6 votes
1 answer
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Space of solutions to a fourth order wave equation

I'm interested in finding solutions a fourth order version of the standard wave equation in $d$ dimensional Minkowski spacetime $\mathcal{M}^d$. Defining $\Box := \partial_0^2 - \sum_{i = 1}^{d-1} \...
Jojo's user avatar
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5 votes
2 answers
260 views

Linear hyperbolic PDE on compact two dimensional domain

Consider the equation $$ \begin{equation} \frac{\partial^2f}{\partial x\partial y}=f \end{equation} $$ on a Jordan domain (i.e. the interior of a simple, closed curve on the plane). The equation is ...
Daniel Castro's user avatar
3 votes
1 answer
185 views

Linear transport equation with Lipschitz conditions

Given the equation here, I would like to ask the following relaxed question: Consider the PDE $$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$ with Schwartz initial data $f(0,x) = ...
Pritam Bemis's user avatar
5 votes
2 answers
347 views

Linear transport equation with unbounded coefficients

Consider the PDE $$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$ with Schwartz initial data $f(0,x) = f_0(x) \in \mathscr S(\mathbb R^n).$ I am wondering then if $q$ and all its ...
Pritam Bemis's user avatar
3 votes
1 answer
262 views

Is this equation of hyperbolic type?

I want to now whether this equation is of hyperbolic type: $$(1-\partial_{xx})y_{tt}+y_{xxxx}=0$$ with boundary conditions $$y(t,0)=y(t,1)=y_x(t,0)=y_x(t,1)=0.$$ I would say that the answer is yes. By ...
Gustave's user avatar
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5 votes
0 answers
196 views

Where to locate $0\in \Omega$ to get $u_{\varepsilon}(0)\neq 0$ where $\Delta u_{\varepsilon} + (\lambda-\varepsilon) u_{\varepsilon} = \frac{1}{|x|}$

Let $\Omega \subset \mathbb{R}^3$ a smooth bounded domain with $0\in \Omega$ and $u_\varepsilon(x)$ the solution to $$ \Delta u_\varepsilon + (\lambda-\varepsilon) u_\varepsilon = \frac{1}{|x|}\quad \...
Jonas's user avatar
  • 241
2 votes
1 answer
263 views

Even and odd solutions for the Schrödinger equation

We consider $2a$ - periodic smooth solutions for \begin{eqnarray*} -\Delta u+V(x)\,u=0\qquad\hbox{in}\:[-a,a] \end{eqnarray*} We assume that $V$ is smooth and even (i.e. $V(-x)=V(x)$). We also assume ...
guest61's user avatar
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2 votes
0 answers
94 views

Evolution PDE in dual space : Generalization of a result of Gelfand

The following result is proved in "Generalized functions", Volume 3, Chapter 2.2 by Gelfand : Let $\Phi$ be a Fréchet space with dual space $\Phi'$ endowed with the weak topology. For ...
Desura's user avatar
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1 vote
0 answers
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How can you make a PDE solution stabilize over time?

this is a problem I have been thinking about lately. I tried asking on stack exchange as well but did not find an answer. Suppose I have a simple linear first order PDE of the form: $$au_x+bu_y=0$$ I ...
GSofer's user avatar
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6 votes
0 answers
139 views

Does $f \in L^1([0,T]; S'(\mathbb R^n))$ define a $(1+n)$-dimensional distribution?

Let $f : [0,T] \rightarrow S'(\mathbb R^n)$ be a family of tempered distributions satisfying $$\langle f(t), \phi \rangle \in L^1([0,T])$$ for any Schwartz function $\phi \in S(\mathbb R^n)$. Does $f$ ...
Desura's user avatar
  • 211
2 votes
1 answer
252 views

How to trap a particle without using potential field which is infinity at some point? (quantum physics) If impossible, how to prove it?

As we all know, the wave function of the stationary state a quantum particle trapped in a rigid box (with infinite potential outside the box) cannot have a non-zero value outside the box. So can we ...
owenBBT's user avatar
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1 vote
1 answer
103 views

Relation between the solutions $v_t=Lv$ and $v_t=Av$ if $A$ is a relatively compact perturbation of the linear operator $L$

In a nutshell, here is my question. I read and know about the relation between the spectra of $L$ and $A$ if $A$ is a relatively compact perturbation of $L$. However, for my purpose, I am interested ...
Gateau au fromage's user avatar
2 votes
1 answer
151 views

What rotations are used as a reduction step in Kenig-Ruiz-Sogge's uniform Sobolev estimate?

I think I have understood the bulk of the paper [KRS], but one of the parts I cannot understand is when the authors reduce Theorem 2.1 (p.332) into Proposition 2.1 (p.335). I can understand all the ...
Calvin Khor's user avatar
0 votes
1 answer
96 views

Reference request: Is if possible to estimate the local behaviour of the solution of $\nabla \cdot a(x) \nabla f=g$ via constant coefficients?

Consider the divergence form uniformly elliptic operator $\nabla \cdot a(x) \nabla$ where the coefficient $a$ are smooth and bounded and $D$ is a bounded and smooth domain of $\mathbb R^d$ $$ \begin{...
Kernel's user avatar
  • 456
5 votes
2 answers
383 views

General solution to an ultrahyperbolic PDE

$\DeclareMathOperator\SO{SO}$The following PDE defined on $\mathbb{R}^2$ $$\frac{\partial}{\partial x}\frac{\partial}{\partial y}f(x,y) = 0,$$ has solution $$f(x,y) = g(x) + h(y),$$ where $g,h : \...
Jojo's user avatar
  • 333
6 votes
2 answers
334 views

Vacuum region with positive measure for the Schrödinger equation

Let us consider the free Schrödinger equation $(i\partial_t+\Delta_x)\psi=0$ in $\mathbb{R}_t\times\mathbb{R}_x^d$. I'm trying to understand the structure of the vacuum region $$\Omega(\psi):=\{(t,x)\...
RaffaeleScandone's user avatar
2 votes
1 answer
237 views

Existence of divergence-free unit vector field in conformally rescaled euclidean metric

Question. Let $\Omega \subset \mathbf{R}^2$ be a convex polygonal domain, equipped with a Riemannian metric $g$. Under which conditions on $g$ is there a vector field $X$ in $\Omega$ with $\mathrm{div}...
Leo Moos's user avatar
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2 votes
0 answers
113 views

Divergence-free constraint for a boundary integral equation

Consider the system $$ \begin{cases} \operatorname{curl} \operatorname{curl} \mathbf{u} = 0 \qquad B \cup (\mathbb{R}^3 \setminus \overline{B}) \\ c_1 (\operatorname{curl} \mathbf{u} \times \mathbf{n})...
GaC's user avatar
  • 163
3 votes
1 answer
258 views

References for Green functions of $\nabla \cdot a \nabla$ on a domain with $a \in L^\infty$

I am looking for a reference for basic properties of the Green function for a symmetric, uniformly elliptic operator $\nabla \cdot a \nabla$ where the coefficients $a_{ij}= a_{ji}$ are only assumed to ...
Kernel's user avatar
  • 456
2 votes
0 answers
143 views

Explicit computation of a norm in context of operator-semigroups and differential equations

I am interested in the explicit calculation of the following norm $\vert \cdot \vert$. Let $X$ a Banach space with norm $\Vert \cdot \Vert$ and $(T(t))_{t \geq 0}$ a strongly continuous one-parameter ...
kumquat's user avatar
  • 63
2 votes
1 answer
162 views

The only rotation fields satisfying this PDE are constant

$\newcommand{\div}{\operatorname{div}}$$\newcommand{\SO}{\operatorname{SO(2)}}$$\newcommand{\R}{\operatorname{\mathbb{R}}}$$\newcommand{\bdx}{\partial_x}$$\newcommand{\bdy}{\partial_y}$$\newcommand{\...
Asaf Shachar's user avatar
  • 6,611
2 votes
2 answers
236 views

Heat flow derivative of entropy

In a 1966 paper (Speed of Approach to Equilibrium for Kac's Caricature of a Maxwellian Gas, Arch. Rational Mech. Anal., Vol. 21), McKean seems to suggest that the successive derivatives of entropy $H (...
Dang Zheng's user avatar
3 votes
1 answer
326 views

Neumann/Robin Laplacian semigroup well-known estimate

Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on: $$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \...
Bogdan's user avatar
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1 vote
0 answers
215 views

Fourier Transform; half space baby problem (new)

This question is related to a prior question i asked, see Fourier Transform ; half space elliptic baby problem. Essentially I am asking the same question now but taking a lot more care. So lets ...
Math604's user avatar
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4 votes
2 answers
217 views

Looking for a reference or the procedure on how to solve the parabolic equation with $L^2$-weight

Let $\zeta, u_0\in L^2(\Omega)$, with $\zeta \geq 0$ and $\Omega\subset \Bbb R^d$ open and bounded. \begin{equation}\label{Star-3.7} \begin{cases} \partial_t u -\Delta u + \zeta u=0 &\mbox{ in }\...
Guy Fsone's user avatar
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